The former Hartlepool shot-stopper raced off his line in the 32nd minute, slipped trying to clear Richard Hope's innocuous long ball and there was Eifion Williams to score his first-ever goal for the visitors.
All Williams had to do was head home into an empty net as Provett tried to scramble back to clear the danger.
It was the first time Bury had lost to Wrexham at Gigg Lane since May 2003 when they were beaten 3-0, and also the Shakers' first loss in their last five meetings with each other.
Boss Chris Casper made one change to the side which beat Exeter City with a late goal in the second round of the FA Cup on Saturday.
Striker Andy Mangan was dropped to the substitutes bench, with Lee Bullock handed the chance to shine in a three-man midfield as the home side looked to avoid a second successive League Two defeat.
A one-minute silence was held before the match in respect for former manager Les Shannon, who died recently aged 81.
Shannon managed the Shakers between 1966-1969 and during that time guided the club to a memorable promotion.
But the current crop of Bury players were unable to grab a victory, one which would have been fitting for the club's ex-boss.
The first chance of the game fell to Neil Roberts on the edge of the area, but his powerful left-foot shot was charged down and the home side cleared the danger.
In the 19th minute Wes Baynes saw a right-foot shot deflected wide for a corner, before the hosts had their best chance of the half.
Nicky Adams saw a shot blocked in 28th minute, the ball fell to midfielder Paul Scott and he forced Wrexham goalkeeper Anthony Williams - who had a loan spell with Bury - to save well, although the nearside linesman had already raised his flag for offside.
Wrexham then hit back to take the lead after a bizarre mistake from keeper Provett. Hope's long ball looked innocuous, but in trying to clear the Bury number one slipped on his backside and gifted Williams the chance to head into an empty net.
However, the goal came at a cost as Wrexham lost their goalscorer moments later with a suspected hamstring injury.
Adams fired wide with a long-range effort just before half-time, while defender Ben Futcher made a superb last-ditch tackle to block substitute Williams' right-foot shot.
In the 69th minute Williams forced Provett to make a smart stop, but it was the keeper's early error which was the difference at the final whistle.